Hydrodynamic and concentration boundary layer

Fig. 10.1. Comparison of hydrodynamic and concentration boundary layer thicknesses. An aqueous flow (i/= 1.0 x 10 cm s 1 for water, D = I x I0 5cm2s-1) at lOcms-1 is incident on the edge of a plate with an embedded plane electrode. The electrode leading edge is either coincident with the plate edge or set I cm back. The concentration boundary layer thickness is calculated using the Llveque approximation.

MHD theory [9-14] has been applied extensively [e.g. 15-21] in conjunction with convective diffusion theory [22-26] to the analysis of external magnetic field effects in the hydrodynamic and concentration boundary layer existing at the electrode/electrolyte interface [2,5,27]. 

Preparatory work for the steps in the scaling up of the membrane reactors has been presented in the previous sections. Now, to maintain the similarity of the membrane reactors between the laboratory and pilot plant, dimensional analysis with a number of dimensionless numbers is introduced in the scaling-up process. Traditionally, the scaling-up of hydrodynamic systems is performed with the aid of dimensionless parameters, which must be kept equal at all scales to be hydrodynamically similar. Dimensional analysis allows one to reduce the number of variables that have to be taken into accoimt for mass transfer determination. For mass transfer under forced convection, there are at least three dimensionless groups the Sherwood number, Sh, which contains the mass transfer coefficient the Reynolds number. Re, which contains the flow velocity and defines the flow condition (laminar/turbulent) and the Schmidt number, Sc, which characterizes the diffusive and viscous properties of the respective fluid and describes the relative extension of the fluid-dynamic and concentration boundary layer. The dependence of Sh on Re, Sc, the characteristic length, Dq/L, and D /L can be described in the form of the power series as shown in Eqn (14.38), in which Dc/a is the gap between cathode and anode Dw/C is gap between reactor wall and cathode, and L is the length of the electrode (Pak. Chung, Ju, 2001) ... 

The convection mass transport of species i may also take place if there exists a bulk fluid motion. The convection mass transfer is analogous to convection heat transfer and occurs between a moving mixture of fluid species and an exposed solid surface. Like hydrodynamic and thermal boundary layers, a concentration boundary layer forms over the surface if the free stream concentration of a species i, differs from species concentration at the surface, Qs, in an external flow over a solid surface as demonstrated in Figure 6.13. 

The concentration polarization occurring in electrodialysis, that is, the concentration profiles at the membrane surface can be calculated by a mass balance taking into account all fluxes in the boundary layer and the hydrodynamic conditions in the flow channel between the membranes. To a first approximation the salt concentration at the membrane surface can be calculated and related to the current density by applying the so-called Nernst film model, which assumes that the bulk solution between the laminar boundary layers has a uniform concentration, whereas the concentration in the boundary layers changes over the thickness of the boundary layer. However, the concentration at the membrane surface and the boundary layer thickness are constant along the flow channel from the cell entrance to the exit. In a practical electrodialysis stack there will be entrance and exit effects and concentration... 

The value of the concentration modulus depends on the convective velocity and the mass-transfer coefficient of the concentration boundary layer (D/ i) that means that on the membrane structure and the hydrodynamic conditions. If the retention coefficient is equal to 1, then c /ch = exp(Pe). The larger convective velocity (or smaller diffusion coefficient) causes higher concentration polarization on the membrane interface. 

Numerous empirical correlations for the prediction of residual NAPL dissolution have been presented in the literature and have been compiled by Khachikian and Harmon [68]. On the other hand, just a few correlations for the rate of interface mass transfer from single-component NAPL pools in saturated, homogeneous porous media have been established, and they are based on numerically determined mass transfer coefficients [69, 70]. These correlations relate a dimensionless mass transfer coefficient, i.e., Sherwood number, to appropriate Peclet numbers, as dictated by dimensional analysis with application of the Buckingham Pi theorem [71,72], and they have been developed under the assumption that the thickness of the concentration boundary layer originating from a dissolving NAPL pool is mainly controlled by the contact time of groundwater with the NAPL-water interface that is directly affected by the interstitial groundwater velocity, hydrodynamic dispersion, and pool size. For uniform... 

Fig. 10.3. Relaxation times for concentration boundary layer. t and hydrodynamic boundary layer, t , calculated for an aqueous flow (i/= 1.0 x 10 Jcnrs-1, D = 1 x 10 cm2 s 1) at 10cm s 1 impinging on the edge of a plate with an embedded electrode set 1 cm back from the plate edge.

Fig. 10.5(a). Transfer function, H, relating modulated current response to modulated flow rate for tube electrode (Reference [20]), rectangular electrode embedded in a wall (Reference [12]) and modulated RDE (amplitude only). The dimensionless modulation frequency see text) is the ratio of the time scale for diffusion across the concentration boundary layer to the timescale for modulation of the hydrodynamics. 

A rather simple interpretation of the behaviour of vibrating electrodes can be obtained by considering the response to a square-wave motion, to which a sinusoid rather crudely approximates [33]. Here, it is considered that the concentration boundary layer is periodically renewed by the instantaneous rapid motion and that in the intervals between the square-wave steps the solution is at rest. This is a reasonable approximation for most practical purposes because the hydrodynamic boundary layer relaxation time is short, (Section 10.3.3). In this simple model, the waveform would instantaneously rise to a limit during the motion, decaying as a function of t m during the static phase. This decay rate will obviously be dependent on the size and geometry of the electrode wire, microwire, band or microband. If the delay time between steps were r then the mean current would vary as (l/r,)/o f 1/2df, i.e., as t, i/2 or as fm. 

This review has attempted to put hydrodynamic modulation methods for electroanalysis and for the study of electrochemical reactions into context with other electrochemical techniques. HM is particularly useful for the extension of detection limits in analysis and for the detection of heterogeneity on electrode surfaces. The timescale addressable using HM methodology is limited by the time taken for diffusion across the concentration boundary layer, typically >0.1 s for conventional RDE and channel electrode geometries. This has meant a restriction on the application of HM to deduce fast reaction mechanisms. New methodologies, employing smaller electrodes and thin layer geometries look to lift this restraint. 

The Schmidt number for the mass transfer is analogous to the Prandtl number for heat transfer. Its physical implication means the relative thickness of the hydrodynamic layer and mass-transfer boundary layer. The ratio of the velocity boundary layer (S) to concentration boundary layer (Sc) is governed by the Schmidt number. The relationship is given by... 

A unique interaction between fluid mechanics and transport exists for filtration processes. Such processes perform better than expected based on the predicted impact of concentration boundary layers. The improvement in performance, a rare occurrence for membrane processes, arises from a combination of hydrodynamic diffusion and inertial lift [51]. Hydrodynamic interactions between particles or colloids that accumulate in the concentration boundary layer lead to shear-induced diffusion away from the membrane surface. Shear-induced diffusion can be significantly larger than molecular diffusion and thereby reduce surface concentrations. For sufficiently large particles at high shear rates, inertial lift becomes the dominant mechanism for particle movement away from the membrane. 

The appearance of boundary layers in pervaporation could be highlighted. These phenomena of polarizations of temperature and concentration influence the transfer during pervaporation. The amplitude of these two polarizations varies according to operational hydrodynamic conditions the boundary layers of polarization decrease with the increase stirring rate. 

In Section 3.IOC an exact solution was obtained for the hydrodynamic boundary layer for isothermal laminar flow past a plate and in Section 5.7A an extension of the Blasius solution was also used to derive an expression for convective heat transfer. In an analogous manner we use the Blasius solution for convective mass transfer for the same geometry and laminar flow. In Fig. 7.9-1 the concentration boundary layer is shown where the concentration of the fluid approaching the plate is and in the fluid adjacent to the surface. 

In Section 3.10 an approximate integral analysis was made for the laminar hydrodynamic and also for the turbulent hydrodynamic boundary layer. This was also done in Section 5.7 for the thermal boundary layer. This approximate integral analysis can also be done in exactly the same manner for the laminar and turbulent concentration boundary layers. 

Consider the electrolyte hydrodynamic flow condition shown in Figure 7.17. This is amass transfer found in electrowinning and electrorefining cells, in which the electrolyte motion is upwards on the anode surface due to the generation of oxygen bubbles, which enhances the mass transfer. The reader can observe in this model that the electrolyte motion is more intense at the top than at the bottom of the electrode as indicated by the arrows. If the current flows, then the electrolyte motion increases and descends to the bottom of the cathode. This is the case in which a significant convective molar flux is superimposed on the Pick s diffusion molar flux. The concentration gradient in the fluid adjacent to the vertical electrode (plate) surface causes a variation in the fluid density and the boundary layer (tf ) develops upwards from laminar to turbulent conditions [2,7]. 

For internal flows, concentration boundary layers develop from both top and bottom surfaces and develop into two regions as shown in the figure concentration entry length and concentration fully developed regions similar to hydrodynamic internal flow as shown in Figure 6.14. 

Reiss and Hanratty (1963) developed a model to obtain the velocity gradient Sw, in the vicinity of a wall of rectangular or cylindrical shape. By using a microelectrode mounted flush with a tube wall, the apparent mass transfer coefficient Ka could be obtained by electrochemical technique. This model was established in the cases where the thickness of the concentration boundary layer (5c) was less than that of the hydrodynamic boundary layer (5v) (figure 1) and was verified in cases of high Schmidt number. 

The phenomenon of concentration polarization, which is observed frequently in membrane separation processes, can be described in mathematical terms, as shown in Figure 30 (71). The usual model, which is weU founded in fluid hydrodynamics, assumes the bulk solution to be turbulent, but adjacent to the membrane surface there exists a stagnant laminar boundary layer of thickness (5) typically 50—200 p.m, in which there is no turbulent mixing. The concentration of the macromolecules in the bulk solution concentration is c,. and the concentration of macromolecules at the membrane surface is c. 

External mass tran.sfer between the external surfaces of the adsorbent particles and the surrounding fluid phase. The driving force is the concentration difference across the boundary layer that surrounds each particle, and the latter is affected by the hydrodynamic conditions outside the particles. 

Open-channel monoliths are better defined. The Sherwood (and Nusselt) number varies mainly in the axial direction due to the formation ofa hydrodynamic boundary layer and a concentration (temperature) boundary layer. Owing to the chemical reactions and heat formation on the surface, the local Sherwood (and Nusselt) numbers depend on the local reaction rate and the reaction rate upstream. A complicating factor is that the traditional Sherwood numbers are usually defined for constant concentration or constant flux on the surface, while, in reahty, the catalytic reaction on the surface exhibits different behavior. 

The hydrodynamic boundary layer has an inner part where the vertical velocity increases to a maximum determined by a balance of viscous and buoyancy forces. In fluids of high Schmidt number, the concentration diffusion layer thickness is of the same order of magnitude as this inner part of the hydrodynamic boundary layer. In the outer part of the hydrodynamic boundary layer, where the vertical velocity decays, the buoyancy force is unimportant. The profile of the vertical velocity component near the electrode can be shown to be parabolic. 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

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